Question

Consider the following unattached spring system. m_(1),m_(1)=1, c_(1),c_(1)=12, m_(2),m_(2)=3 Write the stiffness matrix K=[,-] Find the eigenvalues and eigenvectors of M^(-1)K : Smaller eigenvalue = with eigenvector Larger eigenvalue = with eigenvector If the spring system oscillates beginning with initial displacement u(0)=[[0],[0]] and initial velocity u^(')(0)=[[-15],[1]] then compute the displacements of the masses at time t. u_(1)(t)= u_(2)(t)= Find a nonzero initial velocity vector such that the displacement of the masses will be bounded. u^(')(0)=|~ Consider the following unattached spring system. m,] m=1 c1=12 m2=3 Write the stiffness matrix K Write the matrix M-1K Find the eigenvalues and eigenvectors of M1K Smaller eigenvalue = with eigenvector Larger eigenvalue : with eigenvector If the spring system oscillates beginning with initial displacement u(0) = and initial velocity (0= 0 15 then compute the displacements of the masses at time t. u1(t) u2(t) ind a nonzero initial velocity vector such that the displacement of the masses will be bounded. u0

          Consider the following unattached spring system.
m_(1),m_(1)=1, 
c_(1),c_(1)=12, 
m_(2),m_(2)=3
Write the stiffness matrix K=[,-]
Find the eigenvalues and eigenvectors of M^(-1)K :
Smaller eigenvalue = with eigenvector
Larger eigenvalue = with eigenvector
If the spring system oscillates beginning with initial displacement u(0)=[[0],[0]] and initial velocity u^(')(0)=[[-15],[1]] then compute the displacements of the masses at time t.
u_(1)(t)=
u_(2)(t)=
Find a nonzero initial velocity vector such that the displacement of the masses will be bounded.
u^(')(0)=|~
Consider the following unattached spring system.
m,] m=1 c1=12 m2=3
Write the stiffness matrix K
Write the matrix M-1K
Find the eigenvalues and eigenvectors of M1K
 Smaller eigenvalue =
with eigenvector
 Larger eigenvalue :
with eigenvector
If the spring system oscillates beginning with initial displacement u(0) = and initial velocity (0= 0
15
then compute the displacements of the masses at time t.
u1(t) u2(t)
ind a nonzero initial velocity vector such that the displacement of the masses will be bounded.
u0

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consider the following unattached spring system m1m11 c1c112 m2m23 write the stiffness matrix k find the eigenvalues and eigenvectors of m 1k smaller eigenvalue with eigenvector larger 07181

Added by Darrell W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

07/29/2023

Step 1

Step 1: Write the stiffness matrix K The stiffness matrix K for the unattached spring system is given by: \[ K = \begin{bmatrix} c_1 + c_2 & -c_2 \\ -c_2 & c_2 \end{bmatrix} \] Substitute the given values: \[ K = \begin{bmatrix} 12 & -12 \\ -12 & 12 \end{bmatrix} Show more…

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Consider the following unattached spring system. m_(1),m_(1)=1, c_(1),c_(1)=12, m_(2),m_(2)=3 Write the stiffness matrix K=[,-] Find the eigenvalues and eigenvectors of M^(-1)K : Smaller eigenvalue = with eigenvector Larger eigenvalue = with eigenvector If the spring system oscillates beginning with initial displacement u(0)=[[0],[0]] and initial velocity u^(')(0)=[[-15],[1]] then compute the displacements of the masses at time t. u_(1)(t)= u_(2)(t)= Find a nonzero initial velocity vector such that the displacement of the masses will be bounded. u^(')(0)=|~ Consider the following unattached spring system. m,] m=1 c1=12 m2=3 Write the stiffness matrix K Write the matrix M-1K Find the eigenvalues and eigenvectors of M1K Smaller eigenvalue = with eigenvector Larger eigenvalue : with eigenvector If the spring system oscillates beginning with initial displacement u(0) = and initial velocity (0= 0 15 then compute the displacements of the masses at time t. u1(t) u2(t) ind a nonzero initial velocity vector such that the displacement of the masses will be bounded. u0

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Transcript

00:02 Its part a is as follows.

00:04 The system of equation that describes the motion of the two masses, it can be derived as follows.

00:08 So, this is m1 and x1 double dot is equals to f1 minus f2 and i am giving it equation 1 and this is m2 x2 double dot is equals to f2 plus f3.

00:30 I am giving it equation.

00:31 So, here f1 is the k1 x2 minus x1 minus l1, f2 is equals to k2 multiply with x1 minus x2 plus l2 and f3 is equals to here minus k3 and x2 minus l3 this is also f3 and f3 is equals to k3 multiply with x2 minus l3.

01:11 So, converting the system of two post order equation of the form of this form y dash is equals to ay, you will get the values here.

01:28 So, y1 dash is equals to y3 and y2 dash is equals to y4 and i am writing down here y3 dash is equals to minus k1 divide by m1 multiply with y1 plus k2 y2 divide by m1 minus k2 y1 divide by m1 plus k3 y2 divide by m1 and minus k3 y4 divide by so and y4 dash is equals to k3 y2 divide by m2 minus k3 y4 divide by m2.

02:18 So, here we have given here the conditions.

02:22 So, see y1 is equals to x1, y2 is equals to x2, y3 is equals to x1 double dot and y4 is equals to x2 double dot.

02:38 Now, the coefficient matrix a is the coefficient matrix a will be 0 0 1 0 and 0 0 0 1.

02:52 So, by this we can write down it like this see minus k1 divide by m1 k2 divide by m1 minus k2 plus k3 divide by m1 and k3 divide by m1 and another one will be 0 k3 divide by m2 and 0 and minus k3 divide by m2.

03:27 So, this will be our coefficient matrix a.

03:29 This will be our answer for part a...

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